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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{多元统计分析练习4.1-4.2}
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\date{2024 年 4 月 16 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
设 $x_1,x_2,\cdots,x_n$ 是取自总体 $N(\mu_1,\sigma_1^2)$ 的一个样本，
设 $y_1,y_2,\cdots,y_m$ 是取自总体 $N(\mu_2,\sigma_2^2)$ 的一个样本。
分别在下述两种情况下，求均值差 $\mu_1-\mu_2$ 的置信水平为 $\alpha$ 的置信区间。
\begin{enumerate}
\item  设 $\sigma_1^2$ 和 $\sigma_2^2$ 已知。
\item  设 $\sigma_1^2$ 和 $\sigma_2^2$ 未知但相等。
\end{enumerate}


\vspace{0.2cm}

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\item  %Problem 02
设有 $k$ 个总体 $N(\mu_i,\sigma_i^2)$, 其中 $1\le i\le k$. 对每个 $i$, 
设 $x_{i1},x_{i2},\cdots,x_{in_i}$ 是取自总体 $N(\mu_i,\sigma_i^2)$ 的一个样本。
设这些样本之间也相互独立。对这些总体均值是否相等进行假设检验。

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\item  %Problem 03
设 $x_1,x_2,\cdots,x_n$ 是取自多元正态总体 $N_p(\mu,\Sigma)$ 的一个样本，设 $\Sigma>0$. 
分别在下述两种情况下，进行假设检验
$H_0: \mu=\mu_0, \,\,\, v.s. \,\,\, \mu\neq \mu_0. $
\begin{enumerate}
\item  设 $\Sigma$ 已知。
\item  设 $\Sigma$ 未知。
\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 04
写出自由度为 $n$ 的 $p$ 阶 Wishart 分布的定义。并解释 Wishart 分布与 $\chi^2$ 分布的关系。

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\item  %Problem 05
写出自由度为 $n$ 的 $p$ 阶 Hotelling $T^2$ 分布的定义。并解释 Hotelling 分布与 $t$ 分布的关系。

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\item  %Problem 06
使用R语言计算例子4.2.1. 对某地区农村的6名2周岁男婴的身高、胸围和上半臂围进行测量，得样本数据如表4.2.1所示。根据以往资料，该地区城市2周岁男婴的这三个指标的均值向量为 $\mu_0=(90,58,16)'$. 
设总体是多元正态分布。检验该地区农村男婴的均值是否与城市男婴相同。

\vspace{0.2cm}

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\item  %Problem 07
设 $x_1,x_2,\cdots,x_n$ 是取自多元正态总体 $N_p(\mu,\Sigma)$ 的一个样本，设 $\Sigma>0$. 
求总体均值 $\mu$ 的置信水平为 $\alpha$ 的置信区域。

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\item  %Problem 08
设 $x_1,x_2,\cdots,x_n$ 是取自多元正态总体 $N_p(\mu,\Sigma)$ 的一个样本，设 $\Sigma>0$. 
设 $a\in \mathbb{R}^p$ 是常数向量，记 $y_i=a'x_i$, 则 $y_1,y_2,\cdots,y_n$ 是来自总体 $N_p(a'\mu,a'\Sigma a)$ 的一个样本。
\begin{enumerate}
\item  求 $y_1,y_2,\cdots,y_n$ 的样本均值和方差。
\item  求 $a'\mu$ 的置信度为 $1-\alpha$ 的置信区间。
\item  求一切线性组合 $\{a'\mu \,:\, a\in\mathbb{R}^p\}$ 的总置信度为 $1-\alpha$ 的联合置信区间。
\item  求给定线性组合 $\{a_1'\mu, \cdots, a_k'\mu\}$ 的置信度至少为 $1-\alpha$ 的 Bonferroni 联合置信区间。
\end{enumerate}

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\item  %Problem 09
使用R语言计算例子4.2.2. 为评估某职业培训中心的教学效果，随机抽取8名受训者，进行甲乙两个项目的测试。
其数据列于表格4.2.2. 设总体为二元正态分布。
\begin{enumerate}
\item  求总体均值 $\mu=(\mu_1,\mu_2)'$ 的置信度为 0.90 的置信区域。  
\item  求 $\mu_1$ 和 $\mu_2$ 的置信度为 0.90 的联合置信区间。 
\item  求 $\mu_1$ 和 $\mu_2$ 的置信度为 0.90 的 Bonferroni 联合置信区间。 
\item  画出上述置信区域和联合置信区间的图像。

\end{enumerate}

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\item  %Problem 10
设 $x_1,x_2,\cdots,x_n$ 是取自均值为 $\mu$, 协方差矩阵为 $\Sigma$ 的一个样本，设 $\Sigma>0$. 
设 $n$ 很大，且相对于 $p$ 也很大。使用正态近似，检验假设 $H_0: \mu=\mu_0$.  
写出近似置信区域、近似联合置信区间和近似 Bonferroni 联合置信区间。

\vspace{0.2cm}


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\end{enumerate}


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\end{document}

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